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Non-Chiral S-Matrix of N=4 Super Yang-Mills

Yu-tin Huang

TL;DR

The paper presents a non-chiral formulation of the S-matrix for four-dimensional $\, ext{N}=4$ SYM by using Hatsuda–Siegel projective (half-coset) superspace, showing that the non-chiral dual superconformal symmetry arises as a natural uplift from six dimensions. By mapping dual projective coordinates to on-shell variables $(\,\lambda,\tilde{\lambda},\eta,\bar{\eta})$ and performing half-Fourier transforms, the authors construct explicit non-chiral representations for five-point and general $n$-point MHV and anti-MHV amplitudes, and demonstrate that these data uplift to six dimensions, corresponding to massive 4D amplitudes. They spell out the Yangian structure, with 16 level-1 generators emerging from the non-chiral construction and the rest generated by commutators with the original superconformal algebra. The work also develops ambi-momentum twistor space to express non-chiral amplitudes in a form manifestly cyclic and dual-conformal, and discusses the partial DSC covariance and the role of $y$-coords, pointing toward a full six-dimensional embedding as a future challenge. Overall, the non-chiral approach provides a robust bridge between massless 4D amplitudes, their massive 4D descendants, and fully six-dimensional DSC symmetry, with potential implications for Wilson loop dualities and ABJM-like structures.

Abstract

We discuss the construction of non-chiral S-matrix of four-dimensional N=4 super Yang-Mills using a non-chiral superspace. This construction utilizes the non-chiral representation of dual superconformal symmetry, which is the natural representation from the point of view of the six-dimensional parent theory. The superspace in discussion is projective superspace constructed by Hatsuda and Siegel, and is based on a half coset U(2,2|4)/U(1,1|2)^2_+. We obtain the non-chiral representation of the five-point and general n-point MHV and anti-MHV amplitude. The non-chiral formulation can be straightforwardly lifted to six dimensions, which is equivalent to massive amplitudes in four dimensions.

Non-Chiral S-Matrix of N=4 Super Yang-Mills

TL;DR

The paper presents a non-chiral formulation of the S-matrix for four-dimensional SYM by using Hatsuda–Siegel projective (half-coset) superspace, showing that the non-chiral dual superconformal symmetry arises as a natural uplift from six dimensions. By mapping dual projective coordinates to on-shell variables and performing half-Fourier transforms, the authors construct explicit non-chiral representations for five-point and general -point MHV and anti-MHV amplitudes, and demonstrate that these data uplift to six dimensions, corresponding to massive 4D amplitudes. They spell out the Yangian structure, with 16 level-1 generators emerging from the non-chiral construction and the rest generated by commutators with the original superconformal algebra. The work also develops ambi-momentum twistor space to express non-chiral amplitudes in a form manifestly cyclic and dual-conformal, and discusses the partial DSC covariance and the role of -coords, pointing toward a full six-dimensional embedding as a future challenge. Overall, the non-chiral approach provides a robust bridge between massless 4D amplitudes, their massive 4D descendants, and fully six-dimensional DSC symmetry, with potential implications for Wilson loop dualities and ABJM-like structures.

Abstract

We discuss the construction of non-chiral S-matrix of four-dimensional N=4 super Yang-Mills using a non-chiral superspace. This construction utilizes the non-chiral representation of dual superconformal symmetry, which is the natural representation from the point of view of the six-dimensional parent theory. The superspace in discussion is projective superspace constructed by Hatsuda and Siegel, and is based on a half coset U(2,2|4)/U(1,1|2)^2_+. We obtain the non-chiral representation of the five-point and general n-point MHV and anti-MHV amplitude. The non-chiral formulation can be straightforwardly lifted to six dimensions, which is equivalent to massive amplitudes in four dimensions.

Paper Structure

This paper contains 17 sections, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Projective superspace and superconformal generators.
  3. From projective to on-shell $(\lambda,\tilde{\lambda},\eta,\bar{\eta})$ space.
  4. Generators $\hat{G}_w$:
  5. Generators $\hat{G}_u,\hat{G}_{u'}$:
  6. Generators $\hat{G}_v$:
  7. Non-chiral DSC symmetry as a descendant from six dimensions
  8. Non-chiral amplitudes in dual superspace
  9. Four-dimensional non-chiral amplitudes$\rightarrow$ massive amplitudes
  10. Chiral $\rightarrow$ non-chiral amplitudes
  11. Five-point amplitude in non-chiral superspace
  12. six-point $\overline{\rm MHV}$ amplitude in non-chiral superspace.
  13. The $n$-point ${\rm MHV}-\overline{\rm MHV}$ amplitude in non-chiral superspace.
  14. Non-chiral amplitudes in chiral momentum twistor space
  15. Toward full DCS representation
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The segregation of the DSC generators into level-0 and level-1 Yangian generators in the framework of projective and chiral superspace. The level-1 generators corresponds to the dual $\hat{G}_v$ generators.
  • Figure 2: Extension from the DSC generators to the full Yangian including level-0 and level-1 generators. The green contour encircles the DSC generators, the blue encircles the level-1 Yangian generators and the red contour encircles the level-0 or the original superconformal generators.